When entering the world of curve and surface theory it’s worth remembering there is a lot of jargon, some of it refers to the same things but with preferred names in different fields. You can very quickly end up down a rabbit hole of maths functions.
Non-uniform rational basis spline (NURBS)
http://en.wikipedia.org/wiki/NURBS
Bézier spline is a spline curve
http://en.wikipedia.org/wiki/Spline_(mathematics)
B-spline
Bezier curves are local, as each stretch of curve is specified by four control point and only those four control points. Their main problem is that, while the can easily provide C1 continuity as they allow you to specify the tangents at the endpoints of each stretch of curve, they aren’t good at C2 continuity. That is why b-splines exist; they are designed to give C2 continuity
The Bezier is the curve widely used in 2D applications like Illustrator and Photoshop, and let’s you draw or edit using points connected to handlebars, which are then used to adjust the shape of the curve. Let’s first take a look at a Bezier and note the handles that make it easy to draw and edit. So, Bezier curves, first and foremost, are formula-based for infinite smoothness.
Let’s take a look at the B-spline. Again, this curve is also formula- based, and generates infinite smoothness. However, it doesn’t have handlebars on the curve. It uses a control cage. That cage is outside of the curve. It only touches at the very end points.
NURBS are a generalization of B-Splines.
Here is one difference between them. To apply an affine transformation (e.g. scaling, rotation and translation) to a B-Spline one can apply the transformation on the control points and the transformed B-Spline will then be represented by the new control points. For NURBS this is not only true for affine transformations but also for projective transformations.
NURBS Surfaces
All NURBS surfaces are 4 sided, the end face of a cylinder is a trimmed NURBS surface.
Nurbs v. polygons v. subdivisons
nurbs
NURBS stands for Non-Uniform Rational B-Splines.
• Non-Uniform refers to the parameterization of the curve. Non-Uniform curves allow, among other things, the presence of multi-knots, which are needed to represent Bezier curves.
• Rational refers to the underlying mathematical representation. This property allows NURBS to represent exact conics (such as parabolic curves, circles, and ellipses) in addition to free-form curves.
• B-splines are piecewise polynomial curves that have a parametric representation.
polygons
A polygon is an n-sided shape, defined by its corners (vertices) and the straight lines between them (edges). When you model with polygons you usually use triangles or quadrilaterals (“quads”), although Maya supports polygons with more sides. An individual polygon is often called a face, and is thought of as the filled area defined by its vertices and edges.
subdivisions
Modeling with subdivision surfaces is an easy way to create intricate objects such as human hands. It combines the best features of NURBS and polygonal modeling.
Subdivision surfaces allow you to use a single surface to model complex shapes. A single subdivision surface can have different levels of detail in different regions. That is, a region that has a complex shape can have more control points to allow finer detail, while a simple or flat region needs fewer control points.
How it works
Subdivision surfaces get their name from this dividing into regions of greater detail. You start with a base mesh and divide and subdivide regions into finer and finer detail, with each subdivision giving greater control in that area.
You reshape subdivision surfaces by modifying control points at the different levels of the hierarchy. The base mesh (or “level 0” mesh) allows you to reshape large areas of the overall surface. The subdivided levels allow finer control in specific regions of the surface.
CADCAM Resources 